LMFDB - Embedded newform 2016.2.s.x.865.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,2,Mod(289,2016)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2016, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2016.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2016.s (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$16.0978410475$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.1445900625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 9x^{6} + 77x^{4} + 36x^{2} + 16 $ x^8 + 9x^6 + 77x^4 + 36*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			865.1
Root			$0.342371 - 0.593004i$ of defining polynomial
Character	$\chi$	$=$	2016.865
Dual form			2016.2.s.x.289.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.46040 - 2.52950i) q^{5} +(1.80278 - 1.93649i) q^{7} +O(q^{10})$	$q+(-1.46040 - 2.52950i) q^{5} +(1.80278 - 1.93649i) q^{7} +(2.26556 - 3.92407i) q^{11} +5.53113 q^{13} +(2.23607 - 3.87298i) q^{17} +(-0.342371 - 0.593004i) q^{19} +(1.00000 + 1.73205i) q^{23} +(-1.76556 + 3.05805i) q^{25} -4.29029 q^{29} +(-4.03884 + 6.99548i) q^{31} +(-7.53113 - 1.73205i) q^{35} +(2.76556 + 4.79010i) q^{37} -2.73897 q^{41} +3.78739 q^{43} +(5.53113 + 9.58020i) q^{47} +(-0.500000 - 6.98212i) q^{49} +(5.93254 - 10.2755i) q^{53} -13.2346 q^{55} +(0.734436 - 1.27208i) q^{59} +(-5.53113 - 9.58020i) q^{61} +(-8.07769 - 13.9910i) q^{65} +(6.18399 - 10.7110i) q^{67} -7.06226 q^{71} +(-2.76556 + 4.79010i) q^{73} +(-3.51463 - 11.4615i) q^{77} +(8.32914 + 14.4265i) q^{79} -1.46887 q^{83} -13.0623 q^{85} +(-8.76243 - 15.1770i) q^{89} +(9.97138 - 10.7110i) q^{91} +(-1.00000 + 1.73205i) q^{95} -1.46887 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 2 q^{11} + 12 q^{13} + 8 q^{23} + 2 q^{25} - 28 q^{35} + 6 q^{37} + 12 q^{47} - 4 q^{49} + 22 q^{59} - 12 q^{61} + 8 q^{71} - 6 q^{73} - 44 q^{83} - 40 q^{85} - 8 q^{95} - 44 q^{97}+O(q^{100}) $ 8 * q + 2 * q^11 + 12 * q^13 + 8 * q^23 + 2 * q^25 - 28 * q^35 + 6 * q^37 + 12 * q^47 - 4 * q^49 + 22 * q^59 - 12 * q^61 + 8 * q^71 - 6 * q^73 - 44 * q^83 - 40 * q^85 - 8 * q^95 - 44 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.46040	−	2.52950i	−0.653113	−	1.13122i	−0.982363	−	0.186982i	$-0.940129\pi$
$5$	−1.46040	−	2.52950i	−0.653113	−	1.13122i	0.329250	−	0.944243i	$-0.393204\pi$
$6$		0			0
$7$	1.80278	−	1.93649i	0.681385	−	0.731925i
$7$	1.80278	−	1.93649i	0.681385	−	0.731925i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.26556	−	3.92407i	0.683093	−	1.18315i	−0.290939	−	0.956742i	$-0.593968\pi$
$11$	2.26556	−	3.92407i	0.683093	−	1.18315i	0.974032	−	0.226411i	$-0.0726991\pi$
$12$		0			0
$13$	5.53113			1.53406			0.767030	−	0.641612i	$-0.221734\pi$
$13$	5.53113			1.53406			0.767030	+	0.641612i	$0.221734\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.23607	−	3.87298i	0.542326	−	0.939336i	−0.456444	−	0.889752i	$-0.650877\pi$
$17$	2.23607	−	3.87298i	0.542326	−	0.939336i	0.998770	−	0.0495842i	$-0.0157896\pi$
$18$		0			0
$19$	−0.342371	−	0.593004i	−0.0785453	−	0.136044i	0.824077	−	0.566477i	$-0.191694\pi$
$19$	−0.342371	−	0.593004i	−0.0785453	−	0.136044i	−0.902622	+	0.430433i	$0.858361\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.00000	+	1.73205i	0.208514	+	0.361158i	0.951247	−	0.308431i	$-0.0998038\pi$
$23$	1.00000	+	1.73205i	0.208514	+	0.361158i	−0.742732	+	0.669588i	$0.766471\pi$
$24$		0			0
$25$	−1.76556	+	3.05805i	−0.353113	+	0.611609i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.29029			−0.796687			−0.398344	−	0.917236i	$-0.630415\pi$
$29$	−4.29029			−0.796687			−0.398344	+	0.917236i	$0.630415\pi$
$30$		0			0
$31$	−4.03884	+	6.99548i	−0.725398	+	1.25643i	0.233412	+	0.972378i	$0.425011\pi$
$31$	−4.03884	+	6.99548i	−0.725398	+	1.25643i	−0.958810	+	0.284048i	$0.908323\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−7.53113	−	1.73205i	−1.27299	−	0.292770i
$36$		0			0
$37$	2.76556	+	4.79010i	0.454656	+	0.787487i	0.998668	−	0.0515899i	$-0.0164289\pi$
$37$	2.76556	+	4.79010i	0.454656	+	0.787487i	−0.544012	+	0.839077i	$0.683096\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.73897			−0.427755			−0.213877	−	0.976861i	$-0.568609\pi$
$41$	−2.73897			−0.427755			−0.213877	+	0.976861i	$0.568609\pi$
$42$		0			0
$43$	3.78739			0.577572			0.288786	−	0.957394i	$-0.406748\pi$
$43$	3.78739			0.577572			0.288786	+	0.957394i	$0.406748\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.53113	+	9.58020i	0.806798	+	1.39742i	0.915070	+	0.403294i	$0.132135\pi$
$47$	5.53113	+	9.58020i	0.806798	+	1.39742i	−0.108272	+	0.994121i	$0.534532\pi$
$48$		0			0
$49$	−0.500000	−	6.98212i	−0.0714286	−	0.997446i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.93254	−	10.2755i	0.814897	−	1.41144i	−0.0945046	−	0.995524i	$-0.530127\pi$
$53$	5.93254	−	10.2755i	0.814897	−	1.41144i	0.909402	−	0.415919i	$-0.136540\pi$
$54$		0			0
$55$	−13.2346			−1.78455
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.734436	−	1.27208i	0.0956154	−	0.165611i	−0.814250	−	0.580515i	$-0.802851\pi$
$59$	0.734436	−	1.27208i	0.0956154	−	0.165611i	0.909865	+	0.414904i	$0.136185\pi$
$60$		0			0
$61$	−5.53113	−	9.58020i	−0.708188	−	1.22662i	−0.965528	−	0.260298i	$-0.916179\pi$
$61$	−5.53113	−	9.58020i	−0.708188	−	1.22662i	0.257340	−	0.966321i	$-0.417154\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−8.07769	−	13.9910i	−1.00191	−	1.73537i
$66$		0			0
$67$	6.18399	−	10.7110i	0.755495	−	1.30856i	−0.189633	−	0.981855i	$-0.560730\pi$
$67$	6.18399	−	10.7110i	0.755495	−	1.30856i	0.945128	−	0.326700i	$-0.105937\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−7.06226			−0.838136			−0.419068	−	0.907955i	$-0.637643\pi$
$71$	−7.06226			−0.838136			−0.419068	+	0.907955i	$0.637643\pi$
$72$		0			0
$73$	−2.76556	+	4.79010i	−0.323685	+	0.560639i	−0.981245	−	0.192763i	$-0.938255\pi$
$73$	−2.76556	+	4.79010i	−0.323685	+	0.560639i	0.657560	+	0.753402i	$0.271588\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.51463	−	11.4615i	−0.400529	−	1.30616i
$78$		0			0
$79$	8.32914	+	14.4265i	0.937101	+	1.62311i	0.770844	+	0.637024i	$0.219835\pi$
$79$	8.32914	+	14.4265i	0.937101	+	1.62311i	0.166257	+	0.986082i	$0.446832\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.46887			−0.161230			−0.0806148	−	0.996745i	$-0.525688\pi$
$83$	−1.46887			−0.161230			−0.0806148	+	0.996745i	$0.525688\pi$
$84$		0			0
$85$	−13.0623			−1.41680
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−8.76243	−	15.1770i	−0.928816	−	1.60876i	−0.785307	−	0.619107i	$-0.787495\pi$
$89$	−8.76243	−	15.1770i	−0.928816	−	1.60876i	−0.143509	−	0.989649i	$-0.545839\pi$
$90$		0			0
$91$	9.97138	−	10.7110i	1.04529	−	1.12282i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.00000	+	1.73205i	−0.102598	+	0.177705i
$96$		0			0
$97$	−1.46887			−0.149141			−0.0745706	−	0.997216i	$-0.523759\pi$
$97$	−1.46887			−0.149141			−0.0745706	+	0.997216i	$0.523759\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.55133	−	2.68698i	0.154363	−	0.267364i	−0.778464	−	0.627689i	$-0.784001\pi$
$101$	1.55133	−	2.68698i	0.154363	−	0.267364i	0.932827	+	0.360325i	$0.117334\pi$
$102$		0			0
$103$	−1.89370	−	3.27998i	−0.186592	−	0.323186i	0.757520	−	0.652812i	$-0.226411\pi$
$103$	−1.89370	−	3.27998i	−0.186592	−	0.323186i	−0.944112	+	0.329626i	$0.893077\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.26556	+	12.5843i	0.702389	+	1.21657i	0.967626	+	0.252390i	$0.0812164\pi$
$107$	7.26556	+	12.5843i	0.702389	+	1.21657i	−0.265237	+	0.964183i	$0.585450\pi$
$108$		0			0
$109$	−4.76556	+	8.25420i	−0.456458	+	0.790609i	−0.998771	−	0.0495679i	$-0.984216\pi$
$109$	−4.76556	+	8.25420i	−0.456458	+	0.790609i	0.542312	+	0.840177i	$0.317549\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7.57479			0.712576			0.356288	−	0.934376i	$-0.384042\pi$
$113$	7.57479			0.712576			0.356288	+	0.934376i	$0.384042\pi$
$114$		0			0
$115$	2.92081	−	5.05899i	0.272367	−	0.471753i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.46887	−	11.3122i	−0.317991	−	1.03699i
$120$		0			0
$121$	−4.76556	−	8.25420i	−0.433233	−	0.750382i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−4.29029			−0.383735
$126$		0			0
$127$	−15.6525			−1.38893			−0.694466	−	0.719525i	$-0.744359\pi$
$127$	−15.6525			−1.38893			−0.694466	+	0.719525i	$0.744359\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−4.79669	−	8.30812i	−0.419089	−	0.725884i	0.576759	−	0.816914i	$-0.304317\pi$
$131$	−4.79669	−	8.30812i	−0.419089	−	0.725884i	−0.995848	+	0.0910308i	$0.970984\pi$
$132$		0			0
$133$	−1.76556	−	0.406054i	−0.153094	−	0.0352094i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.29029	+	7.43101i	−0.366544	+	0.634874i	−0.989023	−	0.147763i	$-0.952793\pi$
$137$	−4.29029	+	7.43101i	−0.366544	+	0.634874i	0.622478	+	0.782637i	$0.286126\pi$
$138$		0			0
$139$	−4.79319			−0.406553			−0.203277	−	0.979121i	$-0.565159\pi$
$139$	−4.79319			−0.406553			−0.203277	+	0.979121i	$0.565159\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	12.5311	−	21.7046i	1.04791	−	1.81503i
$144$		0			0
$145$	6.26556	+	10.8523i	0.520327	+	0.901232i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4.29029	−	7.43101i	−0.351474	−	0.608772i	0.635034	−	0.772485i	$-0.280986\pi$
$149$	−4.29029	−	7.43101i	−0.351474	−	0.608772i	−0.986508	+	0.163713i	$0.947653\pi$
$150$		0			0
$151$	−2.14515	+	3.71550i	−0.174570	+	0.302363i	−0.940012	−	0.341141i	$-0.889187\pi$
$151$	−2.14515	+	3.71550i	−0.174570	+	0.302363i	0.765443	+	0.643504i	$0.222520\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	23.5934			1.89507
$156$		0			0
$157$	5.53113	−	9.58020i	0.441432	−	0.764583i	−0.556364	−	0.830939i	$-0.687804\pi$
$157$	5.53113	−	9.58020i	0.441432	−	0.764583i	0.997796	+	0.0663559i	$0.0211373\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	5.15688	+	1.18601i	0.406419	+	0.0934705i
$162$		0			0
$163$	11.8651	+	20.5509i	0.929345	+	1.60967i	0.784420	+	0.620230i	$0.212961\pi$
$163$	11.8651	+	20.5509i	0.929345	+	1.60967i	0.144925	+	0.989443i	$0.453706\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−14.0000			−1.08335			−0.541676	−	0.840587i	$-0.682210\pi$
$167$	−14.0000			−1.08335			−0.541676	+	0.840587i	$0.682210\pi$
$168$		0			0
$169$	17.5934			1.35334
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.60555	+	6.24500i	0.274125	+	0.474798i	0.969914	−	0.243448i	$-0.0782785\pi$
$173$	3.60555	+	6.24500i	0.274125	+	0.474798i	−0.695789	+	0.718246i	$0.744945\pi$
$174$		0			0
$175$	2.73897	+	8.93197i	0.207046	+	0.675194i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−9.53113	+	16.5084i	−0.712390	+	1.23390i	0.251568	+	0.967840i	$0.419054\pi$
$179$	−9.53113	+	16.5084i	−0.712390	+	1.23390i	−0.963958	+	0.266056i	$0.914279\pi$
$180$		0			0
$181$	8.46887			0.629486			0.314743	−	0.949177i	$-0.398082\pi$
$181$	8.46887			0.629486			0.314743	+	0.949177i	$0.398082\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	8.07769	−	13.9910i	0.593883	−	1.02864i
$186$		0			0
$187$	−10.1319	−	17.5490i	−0.740919	−	1.28331i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.46887	+	4.27621i	0.178641	+	0.309416i	0.941415	−	0.337249i	$-0.109497\pi$
$191$	2.46887	+	4.27621i	0.178641	+	0.309416i	−0.762774	+	0.646665i	$0.776163\pi$
$192$		0			0
$193$	9.03113	−	15.6424i	0.650075	−	1.12596i	−0.333029	−	0.942916i	$-0.608071\pi$
$193$	9.03113	−	15.6424i	0.650075	−	1.12596i	0.983104	−	0.183046i	$-0.0585958\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−16.1554			−1.15102			−0.575511	−	0.817794i	$-0.695197\pi$
$197$	−16.1554			−1.15102			−0.575511	+	0.817794i	$0.695197\pi$
$198$		0			0
$199$	−7.21110	+	12.4900i	−0.511182	+	0.885392i	0.488735	+	0.872433i	$0.337459\pi$
$199$	−7.21110	+	12.4900i	−0.511182	+	0.885392i	−0.999916	+	0.0129598i	$0.995875\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−7.73444	+	8.30812i	−0.542851	+	0.583115i
$204$		0			0
$205$	4.00000	+	6.92820i	0.279372	+	0.483887i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−3.10265			−0.214615
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−5.53113	−	9.58020i	−0.377220	−	0.653364i
$216$		0			0
$217$	6.26556	+	20.4325i	0.425334	+	1.38705i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	12.3680	−	21.4220i	0.831960	−	1.44100i
$222$		0			0
$223$	−10.4956			−0.702837			−0.351419	−	0.936218i	$-0.614301\pi$
$223$	−10.4956			−0.702837			−0.351419	+	0.936218i	$0.614301\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.73444	−	13.3964i	0.513353	−	0.889153i	−0.486528	−	0.873665i	$-0.661737\pi$
$227$	7.73444	−	13.3964i	0.513353	−	0.889153i	0.999880	−	0.0154874i	$-0.00492999\pi$
$228$		0			0
$229$	6.82782	+	11.8261i	0.451195	+	0.781493i	0.998461	−	0.0554661i	$-0.0176645\pi$
$229$	6.82782	+	11.8261i	0.451195	+	0.781493i	−0.547265	+	0.836959i	$0.684331\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	8.07769	+	13.9910i	0.529187	+	0.916579i	0.999421	+	0.0340366i	$0.0108363\pi$
$233$	8.07769	+	13.9910i	0.529187	+	0.916579i	−0.470234	+	0.882542i	$0.655830\pi$
$234$		0			0
$235$	16.1554	−	27.9819i	1.05386	−	1.82534i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	21.0623			1.36240			0.681202	−	0.732095i	$-0.261457\pi$
$239$	21.0623			1.36240			0.681202	+	0.732095i	$0.261457\pi$
$240$		0			0
$241$	0.734436	−	1.27208i	0.0473092	−	0.0819419i	−0.841401	−	0.540411i	$-0.818269\pi$
$241$	0.734436	−	1.27208i	0.0473092	−	0.0819419i	0.888710	+	0.458469i	$0.151602\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−16.9310	+	11.4615i	−1.08168	+	0.732246i
$246$		0			0
$247$	−1.89370	−	3.27998i	−0.120493	−	0.208700i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−23.5934			−1.48920			−0.744601	−	0.667510i	$-0.767360\pi$
$251$	−23.5934			−1.48920			−0.744601	+	0.667510i	$0.767360\pi$
$252$		0			0
$253$	9.06226			0.569739
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.60555	−	6.24500i	−0.224908	−	0.389552i	0.731384	−	0.681966i	$-0.238875\pi$
$257$	−3.60555	−	6.24500i	−0.224908	−	0.389552i	−0.956292	+	0.292414i	$0.905542\pi$
$258$		0			0
$259$	14.2617	+	3.27998i	0.886178	+	0.203808i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−15.0623	+	26.0886i	−0.928779	+	1.60869i	−0.143410	+	0.989663i	$0.545807\pi$
$263$	−15.0623	+	26.0886i	−0.928779	+	1.60869i	−0.785369	+	0.619028i	$0.787527\pi$
$264$		0			0
$265$	−34.6556			−2.12888
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−8.16861	+	14.1484i	−0.498049	+	0.862646i	−0.999997	−	0.00225151i	$-0.999283\pi$
$269$	−8.16861	+	14.1484i	−0.498049	+	0.862646i	0.501949	+	0.864897i	$0.332617\pi$
$270$		0			0
$271$	6.43544	+	11.1465i	0.390925	+	0.677102i	0.992572	−	0.121660i	$-0.0388218\pi$
$271$	6.43544	+	11.1465i	0.390925	+	0.677102i	−0.601647	+	0.798762i	$0.705488\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	8.00000	+	13.8564i	0.482418	+	0.835573i
$276$		0			0
$277$	6.82782	−	11.8261i	0.410244	−	0.710564i	−0.584672	−	0.811270i	$-0.698777\pi$
$277$	6.82782	−	11.8261i	0.410244	−	0.710564i	0.994916	+	0.100706i	$0.0321102\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.00580			0.0600008			0.0300004	−	0.999550i	$-0.490449\pi$
$281$	1.00580			0.0600008			0.0300004	+	0.999550i	$0.490449\pi$
$282$		0			0
$283$	7.05057	−	12.2120i	0.419113	−	0.725925i	−0.576737	−	0.816930i	$-0.695674\pi$
$283$	7.05057	−	12.2120i	0.419113	−	0.725925i	0.995850	+	0.0910044i	$0.0290077\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4.93774	+	5.30399i	−0.291466	+	0.313084i
$288$		0			0
$289$	−1.50000	−	2.59808i	−0.0882353	−	0.152828i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	30.7594			1.79698			0.898492	−	0.438989i	$-0.144663\pi$
$293$	30.7594			1.79698			0.898492	+	0.438989i	$0.144663\pi$
$294$		0			0
$295$	−4.29029			−0.249791
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.53113	+	9.58020i	0.319873	+	0.554037i
$300$		0			0
$301$	6.82782	−	7.33426i	0.393549	−	0.422740i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−16.1554	+	27.9819i	−0.925054	+	1.60224i
$306$		0			0
$307$	13.7375			0.784038			0.392019	−	0.919957i	$-0.371777\pi$
$307$	13.7375			0.784038			0.392019	+	0.919957i	$0.371777\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	14.0000	−	24.2487i	0.793867	−	1.37502i	−0.129689	−	0.991555i	$-0.541398\pi$
$311$	14.0000	−	24.2487i	0.793867	−	1.37502i	0.923556	−	0.383464i	$-0.125269\pi$
$312$		0			0
$313$	−2.03113	−	3.51802i	−0.114806	−	0.198850i	0.802896	−	0.596119i	$-0.203291\pi$
$313$	−2.03113	−	3.51802i	−0.114806	−	0.198850i	−0.917702	+	0.397269i	$0.869958\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.7257	+	18.5775i	0.602417	+	1.04342i	0.992454	+	0.122618i	$0.0391289\pi$
$317$	10.7257	+	18.5775i	0.602417	+	1.04342i	−0.390037	+	0.920799i	$0.627538\pi$
$318$		0			0
$319$	−9.71994	+	16.8354i	−0.544212	+	0.942603i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−3.06226			−0.170389
$324$		0			0
$325$	−9.76556	+	16.9145i	−0.541696	+	0.938245i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	28.5234	+	6.55996i	1.57254	+	0.361662i
$330$		0			0
$331$	−14.2617	−	24.7019i	−0.783893	−	1.35774i	−0.929658	−	0.368423i	$-0.879898\pi$
$331$	−14.2617	−	24.7019i	−0.783893	−	1.35774i	0.145766	−	0.989319i	$-0.453435\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−36.1245			−1.97369
$336$		0			0
$337$	−13.1245			−0.714938			−0.357469	−	0.933925i	$-0.616360\pi$
$337$	−13.1245			−0.714938			−0.357469	+	0.933925i	$0.616360\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	18.3005	+	31.6974i	0.991029	+	1.71651i
$342$		0			0
$343$	−14.4222	−	11.6190i	−0.778726	−	0.627364i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.53113	−	2.65199i	0.0821953	−	0.142366i	−0.821997	−	0.569491i	$-0.807140\pi$
$347$	1.53113	−	2.65199i	0.0821953	−	0.142366i	0.904193	+	0.427125i	$0.140474\pi$
$348$		0			0
$349$	−8.12452			−0.434895			−0.217448	−	0.976072i	$-0.569773\pi$
$349$	−8.12452			−0.434895			−0.217448	+	0.976072i	$0.569773\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−14.1011	+	24.4239i	−0.750528	+	1.29995i	0.197038	+	0.980396i	$0.436868\pi$
$353$	−14.1011	+	24.4239i	−0.750528	+	1.29995i	−0.947567	+	0.319558i	$0.896466\pi$
$354$		0			0
$355$	10.3138	+	17.8639i	0.547397	+	0.948120i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	18.5311	+	32.0969i	0.978036	+	1.69401i	0.669530	+	0.742785i	$0.266495\pi$
$359$	18.5311	+	32.0969i	0.978036	+	1.69401i	0.308505	+	0.951223i	$0.400171\pi$
$360$		0			0
$361$	9.26556	−	16.0484i	0.487661	−	0.844654i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	16.1554			0.845611
$366$		0			0
$367$	−4.90543	+	8.49645i	−0.256061	+	0.443511i	−0.965183	−	0.261575i	$-0.915758\pi$
$367$	−4.90543	+	8.49645i	−0.256061	+	0.443511i	0.709122	+	0.705086i	$0.249092\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−9.20331	−	30.0127i	−0.477812	−	1.55818i
$372$		0			0
$373$	−10.8278	−	18.7543i	−0.560643	−	0.971063i	−0.997440	−	0.0715027i	$-0.977221\pi$
$373$	−10.8278	−	18.7543i	−0.560643	−	0.971063i	0.436797	−	0.899560i	$-0.356113\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−23.7302			−1.22217
$378$		0			0
$379$	11.3622			0.583636			0.291818	−	0.956474i	$-0.405740\pi$
$379$	11.3622			0.583636			0.291818	+	0.956474i	$0.405740\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7.00000	+	12.1244i	0.357683	+	0.619526i	0.987573	−	0.157159i	$-0.0502334\pi$
$383$	7.00000	+	12.1244i	0.357683	+	0.619526i	−0.629890	+	0.776684i	$0.716900\pi$
$384$		0			0
$385$	−23.8590	+	25.6286i	−1.21596	+	1.30616i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−15.6525	+	27.1109i	−0.793612	+	1.37458i	0.130105	+	0.991500i	$0.458469\pi$
$389$	−15.6525	+	27.1109i	−0.793612	+	1.37458i	−0.923717	+	0.383076i	$0.874865\pi$
$390$		0			0
$391$	8.94427			0.452331
$392$		0			0
$393$		0			0
$394$		0			0
$395$	24.3278	−	42.1370i	1.22407	−	2.12014i
$396$		0			0
$397$	−9.76556	−	16.9145i	−0.490120	−	0.848912i	0.509816	−	0.860284i	$-0.329714\pi$
$397$	−9.76556	−	16.9145i	−0.490120	−	0.848912i	−0.999935	+	0.0113715i	$0.996380\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.78739	+	6.55996i	0.189133	+	0.327589i	0.944962	−	0.327181i	$-0.106099\pi$
$401$	3.78739	+	6.55996i	0.189133	+	0.327589i	−0.755828	+	0.654770i	$0.772765\pi$
$402$		0			0
$403$	−22.3394	+	38.6929i	−1.11280	+	1.92743i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	25.0623			1.24229
$408$		0			0
$409$	13.0934	−	22.6784i	0.647426	−	1.12138i	−0.336309	−	0.941752i	$-0.609179\pi$
$409$	13.0934	−	22.6784i	0.647426	−	1.12138i	0.983735	−	0.179624i	$-0.0574880\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.13935	−	3.71550i	−0.0560637	−	0.182828i
$414$		0			0
$415$	2.14515	+	3.71550i	0.105301	+	0.182387i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−5.87548			−0.287036			−0.143518	−	0.989648i	$-0.545842\pi$
$419$	−5.87548			−0.287036			−0.143518	+	0.989648i	$0.545842\pi$
$420$		0			0
$421$	2.59339			0.126394			0.0631970	−	0.998001i	$-0.479870\pi$
$421$	2.59339			0.126394			0.0631970	+	0.998001i	$0.479870\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	7.89584	+	13.6760i	0.383005	+	0.663384i
$426$		0			0
$427$	−28.5234	−	6.55996i	−1.38034	−	0.317459i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	6.59339	−	11.4201i	0.317592	−	0.550086i	−0.662393	−	0.749157i	$-0.730459\pi$
$431$	6.59339	−	11.4201i	0.317592	−	0.550086i	0.979985	+	0.199071i	$0.0637923\pi$
$432$		0			0
$433$	−11.4066			−0.548167			−0.274083	−	0.961706i	$-0.588374\pi$
$433$	−11.4066			−0.548167			−0.274083	+	0.961706i	$0.588374\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.684742	−	1.18601i	0.0327556	−	0.0567344i
$438$		0			0
$439$	−7.98677	−	13.8335i	−0.381188	−	0.660236i	0.610045	−	0.792367i	$-0.291151\pi$
$439$	−7.98677	−	13.8335i	−0.381188	−	0.660236i	−0.991232	+	0.132131i	$0.957818\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.32782	−	4.03191i	−0.110598	−	0.191562i	0.805413	−	0.592713i	$-0.201943\pi$
$443$	−2.32782	−	4.03191i	−0.110598	−	0.191562i	−0.916012	+	0.401152i	$0.868610\pi$
$444$		0			0
$445$	−25.5934	+	44.3290i	−1.21324	+	2.10140i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	33.3165			1.57231			0.786153	−	0.618033i	$-0.212070\pi$
$449$	33.3165			1.57231			0.786153	+	0.618033i	$0.212070\pi$
$450$		0			0
$451$	−6.20531	+	10.7479i	−0.292196	+	0.506099i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−41.6556	−	9.58020i	−1.95285	−	0.449127i
$456$		0			0
$457$	−1.96887	−	3.41018i	−0.0920999	−	0.159522i	0.816295	−	0.577636i	$-0.196025\pi$
$457$	−1.96887	−	3.41018i	−0.0920999	−	0.159522i	−0.908395	+	0.418114i	$0.862691\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−9.30796			−0.433515			−0.216757	−	0.976225i	$-0.569548\pi$
$461$	−9.30796			−0.433515			−0.216757	+	0.976225i	$0.569548\pi$
$462$		0			0
$463$	28.5234			1.32559			0.662796	−	0.748800i	$-0.269370\pi$
$463$	28.5234			1.32559			0.662796	+	0.748800i	$0.269370\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5.53113	+	9.58020i	0.255950	+	0.443319i	0.965153	−	0.261686i	$-0.0842784\pi$
$467$	5.53113	+	9.58020i	0.255950	+	0.443319i	−0.709203	+	0.705004i	$0.750945\pi$
$468$		0			0
$469$	−9.59339	−	31.2847i	−0.442981	−	1.44460i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	8.58059	−	14.8620i	0.394536	−	0.683356i
$474$		0			0
$475$	2.41791			0.110941
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2.59339	+	4.49188i	−0.118495	+	0.205239i	−0.919171	−	0.393858i	$-0.871140\pi$
$479$	−2.59339	+	4.49188i	−0.118495	+	0.205239i	0.800677	+	0.599097i	$0.204474\pi$
$480$		0			0
$481$	15.2967	+	26.4947i	0.697469	+	1.20805i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.14515	+	3.71550i	0.0974061	+	0.168712i
$486$		0			0
$487$	11.6136	−	20.1154i	0.526264	−	0.911516i	−0.473268	−	0.880919i	$-0.656926\pi$
$487$	11.6136	−	20.1154i	0.526264	−	0.911516i	0.999532	−	0.0305973i	$-0.00974093\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−10.6556			−0.480882			−0.240441	−	0.970664i	$-0.577292\pi$
$491$	−10.6556			−0.480882			−0.240441	+	0.970664i	$0.577292\pi$
$492$		0			0
$493$	−9.59339	+	16.6162i	−0.432064	+	0.748357i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−12.7317	+	13.6760i	−0.571093	+	0.613453i
$498$		0			0
$499$	−13.7588	−	23.8309i	−0.615928	−	1.06682i	−0.990221	−	0.139507i	$-0.955448\pi$
$499$	−13.7588	−	23.8309i	−0.615928	−	1.06682i	0.374293	−	0.927310i	$-0.377885\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	41.3113			1.84198			0.920990	−	0.389587i	$-0.127382\pi$
$503$	41.3113			1.84198			0.920990	+	0.389587i	$0.127382\pi$
$504$		0			0
$505$	−9.06226			−0.403265
$506$		0			0
$507$		0			0
$508$		0			0
$509$	9.03519	+	15.6494i	0.400478	+	0.693648i	0.993784	−	0.111329i	$-0.0355108\pi$
$509$	9.03519	+	15.6494i	0.400478	+	0.693648i	−0.593306	+	0.804977i	$0.702177\pi$
$510$		0			0
$511$	4.29029	+	13.9910i	0.189791	+	0.618924i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−5.53113	+	9.58020i	−0.243731	+	0.422154i
$516$		0			0
$517$	50.1245			2.20447
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−10.3138	+	17.8639i	−0.451854	+	0.782634i	−0.998501	−	0.0547291i	$-0.982570\pi$
$521$	−10.3138	+	17.8639i	−0.451854	+	0.782634i	0.546647	+	0.837363i	$0.315904\pi$
$522$		0			0
$523$	6.18399	+	10.7110i	0.270407	+	0.468359i	0.968966	−	0.247194i	$-0.0795084\pi$
$523$	6.18399	+	10.7110i	0.270407	+	0.468359i	−0.698559	+	0.715552i	$0.746175\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	18.0623	+	31.2847i	0.786804	+	1.36279i
$528$		0			0
$529$	9.50000	−	16.4545i	0.413043	−	0.715412i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−15.1496			−0.656201
$534$		0			0
$535$	21.2213	−	36.7564i	0.917478	−	1.58912i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−28.5311	−	13.8564i	−1.22892	−	0.596838i
$540$		0			0
$541$	9.76556	+	16.9145i	0.419854	+	0.727209i	0.995924	−	0.0901912i	$-0.0287478\pi$
$541$	9.76556	+	16.9145i	0.419854	+	0.727209i	−0.576070	+	0.817400i	$0.695414\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	27.8386			1.19248
$546$		0			0
$547$	25.7418			1.10064			0.550319	−	0.834954i	$-0.314506\pi$
$547$	25.7418			1.10064			0.550319	+	0.834954i	$0.314506\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.46887	+	2.54416i	0.0625760	+	0.108385i
$552$		0			0
$553$	42.9523	+	9.87842i	1.82652	+	0.420073i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.14515	+	3.71550i	−0.0908928	+	0.157431i	−0.907887	−	0.419215i	$-0.862305\pi$
$557$	−2.14515	+	3.71550i	−0.0908928	+	0.157431i	0.816994	+	0.576646i	$0.195639\pi$
$558$		0			0
$559$	20.9486			0.886030
$560$		0			0
$561$		0			0
$562$		0			0
$563$	14.7344	−	25.5208i	0.620982	−	1.07557i	−0.368321	−	0.929699i	$-0.620067\pi$
$563$	14.7344	−	25.5208i	0.620982	−	1.07557i	0.989303	−	0.145874i	$-0.0465995\pi$
$564$		0			0
$565$	−11.0623	−	19.1604i	−0.465393	−	0.806084i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	20.4457	+	35.4129i	0.857127	+	1.48459i	0.874658	+	0.484741i	$0.161086\pi$
$569$	20.4457	+	35.4129i	0.857127	+	1.48459i	−0.0175307	+	0.999846i	$0.505580\pi$
$570$		0			0
$571$	−10.4743	+	18.1420i	−0.438335	+	0.759219i	−0.997561	−	0.0697965i	$-0.977765\pi$
$571$	−10.4743	+	18.1420i	−0.438335	+	0.759219i	0.559226	+	0.829015i	$0.311098\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−7.06226			−0.294517
$576$		0			0
$577$	10.5000	−	18.1865i	0.437121	−	0.757115i	−0.560345	−	0.828259i	$-0.689332\pi$
$577$	10.5000	−	18.1865i	0.437121	−	0.757115i	0.997466	+	0.0711438i	$0.0226649\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.64805	+	2.84446i	−0.109859	+	0.118008i
$582$		0			0
$583$	−26.8811	−	46.5594i	−1.11330	−	1.92830i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−37.5934			−1.55165			−0.775823	−	0.630951i	$-0.782665\pi$
$587$	−37.5934			−1.55165			−0.775823	+	0.630951i	$0.782665\pi$
$588$		0			0
$589$	5.53113			0.227906
$590$		0			0
$591$		0			0
$592$		0			0
$593$	8.76243	+	15.1770i	0.359830	+	0.623244i	0.987932	−	0.154887i	$-0.0495014\pi$
$593$	8.76243	+	15.1770i	0.359830	+	0.623244i	−0.628102	+	0.778131i	$0.716168\pi$
$594$		0			0
$595$	−23.5483	+	25.2950i	−0.965387	+	1.03699i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	8.53113	−	14.7763i	0.348572	−	0.603745i	−0.637424	−	0.770514i	$-0.720000\pi$
$599$	8.53113	−	14.7763i	0.348572	−	0.603745i	0.985996	+	0.166768i	$0.0533332\pi$
$600$		0			0
$601$	23.2490			0.948348			0.474174	−	0.880431i	$-0.342747\pi$
$601$	23.2490			0.948348			0.474174	+	0.880431i	$0.342747\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−13.9193	+	24.1089i	−0.565900	+	0.980168i
$606$		0			0
$607$	17.4553	+	30.2334i	0.708487	+	1.22714i	0.965418	+	0.260706i	$0.0839552\pi$
$607$	17.4553	+	30.2334i	0.708487	+	1.22714i	−0.256932	+	0.966430i	$0.582711\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	30.5934	+	52.9893i	1.23768	+	2.14372i
$612$		0			0
$613$	−5.00000	+	8.66025i	−0.201948	+	0.349784i	−0.949156	−	0.314806i	$-0.898061\pi$
$613$	−5.00000	+	8.66025i	−0.201948	+	0.349784i	0.747208	+	0.664590i	$0.231394\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−17.1612			−0.690883			−0.345441	−	0.938440i	$-0.612271\pi$
$617$	−17.1612			−0.690883			−0.345441	+	0.938440i	$0.612271\pi$
$618$		0			0
$619$	9.46849	−	16.3999i	0.380571	−	0.659168i	−0.610573	−	0.791960i	$-0.709061\pi$
$619$	9.46849	−	16.3999i	0.380571	−	0.659168i	0.991144	+	0.132792i	$0.0423943\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−45.1868	−	10.3923i	−1.81037	−	0.416359i
$624$		0			0
$625$	15.0934	+	26.1425i	0.603735	+	1.04570i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	24.7360			0.986287
$630$		0			0
$631$	−21.4515			−0.853969			−0.426985	−	0.904259i	$-0.640424\pi$
$631$	−21.4515			−0.853969			−0.426985	+	0.904259i	$0.640424\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	22.8590	+	39.5929i	0.907130	+	1.57119i
$636$		0			0
$637$	−2.76556	−	38.6190i	−0.109576	−	1.53014i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−12.3680	+	21.4220i	−0.488506	+	0.846117i	−0.999913	−	0.0132218i	$-0.995791\pi$
$641$	−12.3680	+	21.4220i	−0.488506	+	0.846117i	0.511407	+	0.859339i	$0.329125\pi$
$642$		0			0
$643$	20.5849			0.811788			0.405894	−	0.913920i	$-0.366960\pi$
$643$	20.5849			0.811788			0.405894	+	0.913920i	$0.366960\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	2.93774	−	5.08832i	0.115495	−	0.200042i	−0.802483	−	0.596675i	$-0.796488\pi$
$647$	2.93774	−	5.08832i	0.115495	−	0.200042i	0.917977	+	0.396633i	$0.129821\pi$
$648$		0			0
$649$	−3.32782	−	5.76396i	−0.130628	−	0.226255i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5.42964	+	9.40442i	0.212478	+	0.368023i	0.952490	−	0.304571i	$-0.0985132\pi$
$653$	5.42964	+	9.40442i	0.212478	+	0.368023i	−0.740011	+	0.672595i	$0.765180\pi$
$654$		0			0
$655$	−14.0102	+	24.2664i	−0.547425	+	0.948168i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	12.9377			0.503983			0.251991	−	0.967729i	$-0.418915\pi$
$659$	12.9377			0.503983			0.251991	+	0.967729i	$0.418915\pi$
$660$		0			0
$661$	−9.76556	+	16.9145i	−0.379836	+	0.657896i	−0.991038	−	0.133579i	$-0.957353\pi$
$661$	−9.76556	+	16.9145i	−0.379836	+	0.657896i	0.611202	+	0.791475i	$0.290686\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.55133	+	5.05899i	0.0601579	+	0.196179i
$666$		0			0
$667$	−4.29029	−	7.43101i	−0.166121	−	0.287730i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−50.1245			−1.93504
$672$		0			0
$673$	9.93774			0.383072			0.191536	−	0.981486i	$-0.438653\pi$
$673$	9.93774			0.383072			0.191536	+	0.981486i	$0.438653\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−12.8226	−	22.2094i	−0.492812	−	0.853576i	0.507154	−	0.861856i	$-0.330698\pi$
$677$	−12.8226	−	22.2094i	−0.492812	−	0.853576i	−0.999966	+	0.00828010i	$0.997364\pi$
$678$		0			0
$679$	−2.64805	+	2.84446i	−0.101623	+	0.109160i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−10.7344	+	18.5926i	−0.410742	+	0.711426i	−0.994971	−	0.100163i	$-0.968064\pi$
$683$	−10.7344	+	18.5926i	−0.410742	+	0.711426i	0.584229	+	0.811589i	$0.301397\pi$
$684$		0			0
$685$	25.0623			0.957580
$686$		0			0
$687$		0			0
$688$		0			0
$689$	32.8136	−	56.8349i	1.25010	−	2.16524i
$690$		0			0
$691$	2.89949	+	5.02207i	0.110302	+	0.191049i	0.915892	−	0.401425i	$-0.131485\pi$
$691$	2.89949	+	5.02207i	0.110302	+	0.191049i	−0.805590	+	0.592473i	$0.798152\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	7.00000	+	12.1244i	0.265525	+	0.459903i
$696$		0			0
$697$	−6.12452	+	10.6080i	−0.231983	+	0.401806i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−11.8651			−0.448138			−0.224069	−	0.974573i	$-0.571934\pi$
$701$	−11.8651			−0.448138			−0.224069	+	0.974573i	$0.571934\pi$
$702$		0			0
$703$	1.89370	−	3.27998i	0.0714221	−	0.123707i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−2.40661	−	7.84815i	−0.0905100	−	0.295160i
$708$		0			0
$709$	19.1245	+	33.1246i	0.718236	+	1.24402i	0.961698	+	0.274111i	$0.0883836\pi$
$709$	19.1245	+	33.1246i	0.718236	+	1.24402i	−0.243462	+	0.969910i	$0.578283\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−16.1554			−0.605024
$714$		0			0
$715$	−73.2021			−2.73760
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−12.5311	−	21.7046i	−0.467332	−	0.809443i	0.531971	−	0.846762i	$-0.321451\pi$
$719$	−12.5311	−	21.7046i	−0.467332	−	0.809443i	−0.999303	+	0.0373193i	$0.988118\pi$
$720$		0			0
$721$	−9.76556	−	2.24594i	−0.363689	−	0.0836431i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	7.57479	−	13.1199i	0.281321	−	0.487262i
$726$		0			0
$727$	30.8021			1.14238			0.571192	−	0.820816i	$-0.306481\pi$
$727$	30.8021			1.14238			0.571192	+	0.820816i	$0.306481\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	8.46887	−	14.6685i	0.313233	−	0.542535i
$732$		0			0
$733$	−4.23444	−	7.33426i	−0.156402	−	0.270897i	0.777166	−	0.629295i	$-0.216656\pi$
$733$	−4.23444	−	7.33426i	−0.156402	−	0.270897i	−0.933569	+	0.358398i	$0.883323\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−28.0205	−	48.5329i	−1.03215	−	1.78773i
$738$		0			0
$739$	−5.17819	+	8.96889i	−0.190483	+	0.329926i	−0.945410	−	0.325882i	$-0.894339\pi$
$739$	−5.17819	+	8.96889i	−0.190483	+	0.329926i	0.754927	+	0.655808i	$0.227672\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	21.0623			0.772699			0.386350	−	0.922352i	$-0.373736\pi$
$743$	21.0623			0.772699			0.386350	+	0.922352i	$0.373736\pi$
$744$		0			0
$745$	−12.5311	+	21.7046i	−0.459105	+	0.795193i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	37.4676	+	8.61701i	1.36904	+	0.314859i
$750$		0			0
$751$	−0.754348	−	1.30657i	−0.0275265	−	0.0476774i	0.851934	−	0.523649i	$-0.175430\pi$
$751$	−0.754348	−	1.30657i	−0.0275265	−	0.0476774i	−0.879461	+	0.475972i	$0.842096\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	12.5311			0.456054
$756$		0			0
$757$	−27.0623			−0.983594			−0.491797	−	0.870710i	$-0.663660\pi$
$757$	−27.0623			−0.983594			−0.491797	+	0.870710i	$0.663660\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−12.1861	−	21.1070i	−0.441747	−	0.765128i	0.556072	−	0.831134i	$-0.312308\pi$
$761$	−12.1861	−	21.1070i	−0.441747	−	0.765128i	−0.997819	+	0.0660056i	$0.978974\pi$
$762$		0			0
$763$	7.39295	+	24.1089i	0.267643	+	0.872802i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4.06226	−	7.03604i	0.146680	−	0.254057i
$768$		0			0
$769$	−46.0623			−1.66105			−0.830524	−	0.556983i	$-0.811959\pi$
$769$	−46.0623			−1.66105			−0.830524	+	0.556983i	$0.811959\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	14.6040	−	25.2950i	0.525271	−	0.909796i	−0.474296	−	0.880366i	$-0.657297\pi$
$773$	14.6040	−	25.2950i	0.525271	−	0.909796i	0.999567	−	0.0294306i	$-0.00936942\pi$
$774$		0			0
$775$	−14.2617	−	24.7019i	−0.512295	−	0.887320i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	0.937742	+	1.62422i	0.0335981	+	0.0581936i
$780$		0			0
$781$	−16.0000	+	27.7128i	−0.572525	+	0.991642i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−32.3107			−1.15322
$786$		0			0
$787$	−24.7360	+	42.8439i	−0.881742	+	1.52722i	−0.0323386	+	0.999477i	$0.510295\pi$
$787$	−24.7360	+	42.8439i	−0.881742	+	1.52722i	−0.849403	+	0.527745i	$0.823038\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	13.6556	−	14.6685i	0.485539	−	0.521552i
$792$		0			0
$793$	−30.5934	−	52.9893i	−1.08640	−	1.88171i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−40.3458			−1.42912			−0.714561	−	0.699573i	$-0.753373\pi$
$797$	−40.3458			−1.42912			−0.714561	+	0.699573i	$0.753373\pi$
$798$		0			0
$799$	49.4719			1.75019
$800$		0			0
$801$		0			0
$802$		0			0
$803$	12.5311	+	21.7046i	0.442214	+	0.765937i
$804$		0			0
$805$	−4.53113	−	14.7763i	−0.159701	−	0.520798i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	12.3680	−	21.4220i	0.434835	−	0.753156i	−0.562447	−	0.826833i	$-0.690140\pi$
$809$	12.3680	−	21.4220i	0.434835	−	0.753156i	0.997282	+	0.0736769i	$0.0234734\pi$
$810$		0			0
$811$	8.58059			0.301305			0.150653	−	0.988587i	$-0.451863\pi$
$811$	8.58059			0.301305			0.150653	+	0.988587i	$0.451863\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	34.6556	−	60.0253i	1.21393	−	2.10260i
$816$		0			0
$817$	−1.29669	−	2.24594i	−0.0453656	−	0.0785754i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−10.2228	−	17.7065i	−0.356779	−	0.617960i	0.630641	−	0.776074i	$-0.282792\pi$
$821$	−10.2228	−	17.7065i	−0.356779	−	0.617960i	−0.987421	+	0.158114i	$0.949459\pi$
$822$		0			0
$823$	16.1554	−	27.9819i	0.563141	−	0.975389i	−0.434079	−	0.900875i	$-0.642926\pi$
$823$	16.1554	−	27.9819i	0.563141	−	0.975389i	0.997220	−	0.0745139i	$-0.0237405\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	2.53113			0.0880160			0.0440080	−	0.999031i	$-0.485987\pi$
$827$	2.53113			0.0880160			0.0440080	+	0.999031i	$0.485987\pi$
$828$		0			0
$829$	19.3590	−	33.5307i	0.672364	−	1.16457i	−0.304868	−	0.952395i	$-0.598612\pi$
$829$	19.3590	−	33.5307i	0.672364	−	1.16457i	0.977232	−	0.212174i	$-0.0680544\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−28.1597	−	13.6760i	−0.975675	−	0.473845i
$834$		0			0
$835$	20.4457	+	35.4129i	0.707551	+	1.22552i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−25.0623			−0.865245			−0.432623	−	0.901575i	$-0.642412\pi$
$839$	−25.0623			−0.865245			−0.432623	+	0.901575i	$0.642412\pi$
$840$		0			0
$841$	−10.5934			−0.365289
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−25.6935	−	44.5024i	−0.883882	−	1.53093i
$846$		0			0
$847$	−24.5754	−	5.65199i	−0.844422	−	0.194205i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−5.53113	+	9.58020i	−0.189605	+	0.328405i
$852$		0			0
$853$	44.5934			1.52685			0.763424	−	0.645897i	$-0.223517\pi$
$853$	44.5934			1.52685			0.763424	+	0.645897i	$0.223517\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−23.0028	+	39.8420i	−0.785760	+	1.36098i	0.142783	+	0.989754i	$0.454395\pi$
$857$	−23.0028	+	39.8420i	−0.785760	+	1.36098i	−0.928544	+	0.371223i	$0.878939\pi$
$858$		0			0
$859$	−17.8885	−	30.9839i	−0.610349	−	1.05716i	−0.991181	−	0.132512i	$-0.957696\pi$
$859$	−17.8885	−	30.9839i	−0.610349	−	1.05716i	0.380832	−	0.924644i	$-0.375638\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−12.6556	−	21.9202i	−0.430803	−	0.746173i	0.566140	−	0.824309i	$-0.308436\pi$
$863$	−12.6556	−	21.9202i	−0.430803	−	0.746173i	−0.996943	+	0.0781365i	$0.975103\pi$
$864$		0			0
$865$	10.5311	−	18.2405i	0.358069	−	0.620194i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	75.4808			2.56051
$870$		0			0
$871$	34.2044	−	59.2438i	1.15897	−	2.00740i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−7.73444	+	8.30812i	−0.261472	+	0.280866i
$876$		0			0
$877$	1.40661	+	2.43633i	0.0474980	+	0.0822689i	0.888797	−	0.458301i	$-0.151542\pi$
$877$	1.40661	+	2.43633i	0.0474980	+	0.0822689i	−0.841299	+	0.540570i	$0.818209\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	34.0439			1.14697			0.573484	−	0.819216i	$-0.305591\pi$
$881$	34.0439			1.14697			0.573484	+	0.819216i	$0.305591\pi$
$882$		0			0
$883$	−18.9370			−0.637280			−0.318640	−	0.947876i	$-0.603226\pi$
$883$	−18.9370			−0.637280			−0.318640	+	0.947876i	$0.603226\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.46887	+	2.54416i	0.0493199	+	0.0854245i	0.889631	−	0.456679i	$-0.150961\pi$
$887$	1.46887	+	2.54416i	0.0493199	+	0.0854245i	−0.840312	+	0.542104i	$0.817628\pi$
$888$		0			0
$889$	−28.2179	+	30.3109i	−0.946398	+	1.01659i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3.78739	−	6.55996i	0.126740	−	0.219521i
$894$		0			0
$895$	55.6772			1.86108
$896$		0			0
$897$		0			0
$898$		0			0
$899$	17.3278	−	30.0127i	0.577915	−	1.00098i
$900$		0			0
$901$	−26.5311	−	45.9533i	−0.883880	−	1.53093i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−12.3680	−	21.4220i	−0.411126	−	0.712090i
$906$		0			0
$907$	−10.9772	+	19.0130i	−0.364491	+	0.631318i	−0.988694	−	0.149945i	$-0.952090\pi$
$907$	−10.9772	+	19.0130i	−0.364491	+	0.631318i	0.624203	+	0.781262i	$0.285424\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	37.1868			1.23205			0.616026	−	0.787726i	$-0.288741\pi$
$911$	37.1868			1.23205			0.616026	+	0.787726i	$0.288741\pi$
$912$		0			0
$913$	−3.32782	+	5.76396i	−0.110135	+	0.190759i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−24.7360	−	5.68891i	−0.816853	−	0.187864i
$918$		0			0
$919$	−14.2617	−	24.7019i	−0.470449	−	0.814842i	0.528980	−	0.848635i	$-0.322575\pi$
$919$	−14.2617	−	24.7019i	−0.470449	−	0.814842i	−0.999429	+	0.0337924i	$0.989241\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−39.0623			−1.28575
$924$		0			0
$925$	−19.5311			−0.642180
$926$		0			0
$927$		0			0
$928$		0			0
$929$	16.8401	+	29.1679i	0.552506	+	0.956969i	0.998093	+	0.0617301i	$0.0196618\pi$
$929$	16.8401	+	29.1679i	0.552506	+	0.956969i	−0.445587	+	0.895239i	$0.647005\pi$
$930$		0			0
$931$	−3.96924	+	2.68698i	−0.130086	+	0.0880621i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−29.5934	+	51.2572i	−0.967807	+	1.67629i
$936$		0			0
$937$	−43.1245			−1.40882			−0.704408	−	0.709795i	$-0.748788\pi$
$937$	−43.1245			−1.40882			−0.704408	+	0.709795i	$0.748788\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	11.9560	−	20.7084i	0.389755	−	0.675075i	−0.602662	−	0.797997i	$-0.705893\pi$
$941$	11.9560	−	20.7084i	0.389755	−	0.675075i	0.992416	+	0.122922i	$0.0392265\pi$
$942$		0			0
$943$	−2.73897	−	4.74403i	−0.0891930	−	0.154487i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	6.00000	+	10.3923i	0.194974	+	0.337705i	0.946892	−	0.321552i	$-0.104204\pi$
$947$	6.00000	+	10.3923i	0.194974	+	0.337705i	−0.751918	+	0.659256i	$0.770871\pi$
$948$		0			0
$949$	−15.2967	+	26.4947i	−0.496552	+	0.860053i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−9.58638			−0.310533			−0.155267	−	0.987873i	$-0.549624\pi$
$953$	−9.58638			−0.310533			−0.155267	+	0.987873i	$0.549624\pi$
$954$		0			0
$955$	7.21110	−	12.4900i	0.233346	−	0.404167i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	6.65564	+	21.7046i	0.214922	+	0.700877i
$960$		0			0
$961$	−17.1245	−	29.6605i	−0.552404	−	0.956791i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−52.7564			−1.69829
$966$		0			0
$967$	15.6525			0.503350			0.251675	−	0.967812i	$-0.419019\pi$
$967$	15.6525			0.503350			0.251675	+	0.967812i	$0.419019\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−22.8590	−	39.5929i	−0.733579	−	1.27060i	−0.955344	−	0.295495i	$-0.904515\pi$
$971$	−22.8590	−	39.5929i	−0.733579	−	1.27060i	0.221765	−	0.975100i	$-0.428818\pi$
$972$		0			0
$973$	−8.64105	+	9.28198i	−0.277019	+	0.297566i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	23.7302	−	41.1018i	0.759195	−	1.31496i	−0.184066	−	0.982914i	$-0.558926\pi$
$977$	23.7302	−	41.1018i	0.759195	−	1.31496i	0.943262	−	0.332051i	$-0.107741\pi$
$978$		0			0
$979$	−79.4074			−2.53787
$980$		0			0
$981$		0			0
$982$		0			0
$983$	19.5311	−	33.8289i	0.622946	−	1.07897i	−0.365988	−	0.930620i	$-0.619269\pi$
$983$	19.5311	−	33.8289i	0.622946	−	1.07897i	0.988934	−	0.148355i	$-0.0473978\pi$
$984$		0			0
$985$	23.5934	+	40.8649i	0.751748	+	1.30207i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	3.78739	+	6.55996i	0.120432	+	0.208595i
$990$		0			0
$991$	12.1165	−	20.9864i	0.384894	−	0.666656i	−0.606860	−	0.794809i	$-0.707571\pi$
$991$	12.1165	−	20.9864i	0.384894	−	0.666656i	0.991755	+	0.128152i	$0.0409046\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	42.1245			1.33544
$996$		0			0
$997$	−11.2344	+	19.4586i	−0.355798	+	0.616260i	−0.987254	−	0.159151i	$-0.949124\pi$
$997$	−11.2344	+	19.4586i	−0.355798	+	0.616260i	0.631456	+	0.775412i	$0.282458\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.2.s.x.865.1	yes	8
3.2	odd	2		2016.2.s.w.865.4	yes	8
4.3	odd	2		2016.2.s.w.865.1	yes	8
7.2	even	3	inner	2016.2.s.x.289.1	yes	8
12.11	even	2	inner	2016.2.s.x.865.4	yes	8
21.2	odd	6		2016.2.s.w.289.4	yes	8
28.23	odd	6		2016.2.s.w.289.1	✓	8
84.23	even	6	inner	2016.2.s.x.289.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2016.2.s.w.289.1	✓	8	28.23	odd	6
2016.2.s.w.289.4	yes	8	21.2	odd	6
2016.2.s.w.865.1	yes	8	4.3	odd	2
2016.2.s.w.865.4	yes	8	3.2	odd	2
2016.2.s.x.289.1	yes	8	7.2	even	3	inner
2016.2.s.x.289.4	yes	8	84.23	even	6	inner
2016.2.s.x.865.1	yes	8	1.1	even	1	trivial
2016.2.s.x.865.4	yes	8	12.11	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2016.s (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.46040 - 2.52950i) q^{5} +(1.80278 - 1.93649i) q^{7} +O(q^{10})\)	\(q+(-1.46040 - 2.52950i) q^{5} +(1.80278 - 1.93649i) q^{7} +(2.26556 - 3.92407i) q^{11} +5.53113 q^{13} +(2.23607 - 3.87298i) q^{17} +(-0.342371 - 0.593004i) q^{19} +(1.00000 + 1.73205i) q^{23} +(-1.76556 + 3.05805i) q^{25} -4.29029 q^{29} +(-4.03884 + 6.99548i) q^{31} +(-7.53113 - 1.73205i) q^{35} +(2.76556 + 4.79010i) q^{37} -2.73897 q^{41} +3.78739 q^{43} +(5.53113 + 9.58020i) q^{47} +(-0.500000 - 6.98212i) q^{49} +(5.93254 - 10.2755i) q^{53} -13.2346 q^{55} +(0.734436 - 1.27208i) q^{59} +(-5.53113 - 9.58020i) q^{61} +(-8.07769 - 13.9910i) q^{65} +(6.18399 - 10.7110i) q^{67} -7.06226 q^{71} +(-2.76556 + 4.79010i) q^{73} +(-3.51463 - 11.4615i) q^{77} +(8.32914 + 14.4265i) q^{79} -1.46887 q^{83} -13.0623 q^{85} +(-8.76243 - 15.1770i) q^{89} +(9.97138 - 10.7110i) q^{91} +(-1.00000 + 1.73205i) q^{95} -1.46887 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 2 q^{11} + 12 q^{13} + 8 q^{23} + 2 q^{25} - 28 q^{35} + 6 q^{37} + 12 q^{47} - 4 q^{49} + 22 q^{59} - 12 q^{61} + 8 q^{71} - 6 q^{73} - 44 q^{83} - 40 q^{85} - 8 q^{95} - 44 q^{97}+O(q^{100}) \) 8 * q + 2 * q^11 + 12 * q^13 + 8 * q^23 + 2 * q^25 - 28 * q^35 + 6 * q^37 + 12 * q^47 - 4 * q^49 + 22 * q^59 - 12 * q^61 + 8 * q^71 - 6 * q^73 - 44 * q^83 - 40 * q^85 - 8 * q^95 - 44 * q^97

Introduction

L-functions

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